Appendix AList of Symbols
The following is a list of symbols used throughout the description of the sticky par-
ticle star formation model (see chapter 3)
αc : Slope of the molecular cloud mass-radius relation. Eq 3.5
ch : Sound speed of the ambient gas phase
ε? : Fraction of a GMC converted to stars in a collapse. E51 : Energy ejected per SnII in
units of 1051ergs
Eb : Total energy in a supernova blast wave
Em : Total kinetic energy in molecular clouds of mass m in a given volume
fcl : Filling factor of cold clouds
fm(σ1, σ2) : Fraction of collisions between clouds with velocity dispersions σ1 and σ2
that lead to mergers
K(m,m′) : The kernel for aggregation of clouds of masses m and m′. Eq. 3.6
λ : Constant of proportionality relating cloud mass and destruction rate by thermal con-
duction. Eq. 3.50
ΛN : Normalised radiative cooling rate
Λnet : Net radiative cooling rate (ergs cm−3s−1)
Mc : Mass of a molecular cloud
Mref : Reference cold cloud mass. Eq 3.5
M?,min : Minimum allowed star mass
M?,max : Maximum allowed star mass
nb : Density internal to a supernova remnant in atoms / cm3 nc : Density of a molecular
cloud in atoms / cm3
nh : Density of the ambient medium in atoms / cm3
NSF : The slope of the schmidt law. Eq 3.36
n(m, t) : The number of clouds with masses between m and m+ dm
N(m, t) : The number density of clouds with masses between m and m+ dm
φ : Efficiency of destruction of cold clouds by thermal conduction
Q : Porosity of the interstellar medium. Sec. 3.2.6
173
A. List of Symbols 174
rc : Radius of a molecular cloud
rref : Reference cold cloud radius. Eq 3.5
rb : The radius of a spherical blast wave
ρc : Mean density of molecular clouds contained in a volume
ρh : Mean density of ambient gas contained in a volume
ρth : Density at which ambient gas becomes thermally unstable
ρSFR : Volume density of star formation
η : Fraction of cloud velocity lost to ’cooling’ collision
Tb : Mean temperature inside of a supernova remnant
Tc : Internal temperature of cold clouds
Th : Temperature of the ambient gas phase
ub : Thermal energy per unit mass of supernova remnants
uc : Thermal energy per unit mass of the cold clouds
uh : Thermal energy per unit mass of the ambient phase
Σ : Cross section for collision between clouds. Eq. 3.7
Σcond : Efficiency of thermal conduction. Eq 3.46
vapp : Relative approach velocity of two molecular clouds
vstick : Maximum relative velocity for cloud merger
x : Slope of the stellar IMF
Appendix BThe Green’s Function ofthe Finite DifferencedLaplacian
For some function φ, defined on a regular grid at points i, with grid spacing ∆ the
finite-difference approximation to the Laplacian at point i is given by
∇2φi ≈φi+1 + φi−1 − 2φi
∆2. (B.1)
We now note that for some function g(x), F(g(t)
)= G(k), where the notation F repre-
sents a Fourier transform, defined as
g(x) =
∫ ∞
∞G(k)e2πikxdk . (B.2)
k represents a frequency, we can write
∇2φ =∑
k
φ(k)ei2πk∆ + φ(k)e−i2πk∆ − 2φ(k)
∆2e2πikx , (B.3)
by using F(g(t − a)
)= e−i2πakG(k). Now noting that eiax = cos(ax) + isin(ax) we can
write
∇2φ =∑
k
φ(k)cos(2πk∆) + isin(2πk∆) + cos(−2πk∆) + isin(−2πk∆) − 2
∆2eikx , (B.4)
which, through symmetry, becomes
∇2φ =∑
k
φ(k)2cos(2πk∆) − 2
∆2e2πikx . (B.5)
and substituting in cos(2x) = 1 − 2sin2(x) we obtain
∇2φ =2
∆2
∑φ(k)sin2(πk∆)e2πikx , (B.6)
which is equal to the right hand side of the Poisson equation. Then we can say that (after
taking a fourier transform)φ(k)
G(k)= ρ(k) (B.7)
175
B. The Green’s Function of the Finite Differenced Laplacian 176
where we have defined the Greens function as
Gj,k,l =
(2
∆2sin2(πk∆)
)−1
(B.8)
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